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Theorem eqer 6074
Description: Equivalence relation involving equality of dependent classes and . (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
eqer.1
eqer.2  R  { <. , 
>.  |  }
Assertion
Ref Expression
eqer  R  Er  _V
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()    R(,)

Proof of Theorem eqer
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqer.2 . . . . 5  R  { <. , 
>.  |  }
21relopabi 4406 . . . 4  Rel  R
32a1i 9 . . 3  Rel  R
4 id 19 . . . . . 6  [_  ]_  [_  ]_  [_  ]_  [_  ]_
54eqcomd 2042 . . . . 5  [_  ]_  [_  ]_  [_  ]_  [_  ]_
6 eqer.1 . . . . . 6
76, 1eqerlem 6073 . . . . 5  R  [_  ]_  [_  ]_
86, 1eqerlem 6073 . . . . 5  R  [_  ]_  [_  ]_
95, 7, 83imtr4i 190 . . . 4  R  R
109adantl 262 . . 3  R  R
11 eqtr 2054 . . . . 5 
[_  ]_  [_  ]_  [_  ]_  [_  ]_  [_  ]_  [_  ]_
126, 1eqerlem 6073 . . . . . 6  R  [_  ]_  [_  ]_
137, 12anbi12i 433 . . . . 5  R  R  [_  ]_  [_  ]_  [_  ]_  [_  ]_
146, 1eqerlem 6073 . . . . 5  R  [_  ]_  [_  ]_
1511, 13, 143imtr4i 190 . . . 4  R  R  R
1615adantl 262 . . 3  R  R  R
17 vex 2554 . . . . 5 
_V
18 eqid 2037 . . . . . 6  [_  ]_  [_  ]_
196, 1eqerlem 6073 . . . . . 6  R  [_  ]_  [_  ]_
2018, 19mpbir 134 . . . . 5  R
2117, 202th 163 . . . 4  _V  R
2221a1i 9 . . 3  _V  R
233, 10, 16, 22iserd 6068 . 2  R  Er  _V
2423trud 1251 1  R  Er  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wtru 1243   wcel 1390   _Vcvv 2551   [_csb 2846   class class class wbr 3755   {copab 3808   Rel wrel 4293    Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042
This theorem is referenced by:  ider  6075
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